Finite Group Databasebeta

← #26122685

Subgroups — #26122685

Order 12032.

916 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689141C211yesmaximal, characteristic
2#20251905C211yesmaximal, characteristic
3C2×#15689096C211yesmaximal, characteristic
4#20251807C211yesmaximal, characteristic, complemented
5#20243098C211yesmaximal, characteristic, complemented
6C2×#15688879C211yesmaximal, characteristic, complemented
7C47×#1740C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11C2×#13093983C2211yescharacteristic
12#15689141—22no—
13#15689141—22no—
14C2×#13093952C414yescomplemented
15#15689131C412yescomplemented
16#15689127C412yescomplemented
17#15689122—22no—
18#15689117—22no—
19#15689096—22no—
20#15689096C414yescomplemented
21C2×#13093938—22no—
22#15689079—22no—
23#15689079—22no—
24C4×#10625329—22no—
25#15688883—22no—
26#15688879—22no—
27C2×#117×C47C2211yescharacteristic
28C2×#117×C47C2211yescharacteristic
29C2×#117×C47C2211yescharacteristic
30C47×#342—22no—
31C2×#10625350—21no—
32C2×#10625350—21no—
33C2×#10625350—21no—
34C2×#10625350—21no—
35C2×#10625350—21no—
36C2×#10625350—21no—
37C2×#10625350—21no—
38C2×#10625350—21no—
39C2×#10625350—21no—
40C2×#10625350—21no—
41C2×#10625350C2×C412yes—
42C2×#10625350C2×C412yes—
43C22×Dic94C2×C411yescharacteristic
44C22×D188C2×C411yescharacteristic
45C22×C22⋊D47—21no—
46C22×C4×D47—21no—
47C2×#10625329—21no—
48C2×#10625329—21no—
49C2×#10625329—21no—
50C2×#10625329—21no—
51C2×#10625329—21no—
52C2×#10625329—21no—
53C2×#10625329—21no—
54C2×#10625329C2×C412yes—
55C2×#10625329C2×C412yes—
56C2×#10625329D412yes—
57C2×#10625329D412yes—
58C2×#10625329D412yes—
59C22×C47⋊C8—21no—
60C22×C47⋊C8D412yes—
61#13093983—21no—
62#13093983—21no—
63#13093983—21no—
64#13093983C2×C412yes—
65C2×C4⋊Dic47—21no—
66C2×C22⋊Dic47—21no—
67C2×C4×Dic47—21no—
68#13093952—28no—
69#13093947—24no—
70#13093944—24no—
71#13093938—28no—
72#13093935—24no—
73C4×C47⋊C8—24no—
74C23×C4×C47C2311yescharacteristic
75#117×C47—21no—
76#117×C47—21no—
77#117×C47—21no—
78#117×C47—21no—
79#117×C47—21no—
80#117×C47—21no—
81#117×C47—21no—
82#117×C47—21no—
83#117×C47—21no—
84#117×C47C2×C412yes—
85#117×C47C2×C412yes—
86#117×C47D412yes—
87C2×C4⋊C4×C47—21no—
88C2×C4⋊C4×C47—21no—
89C2×C4⋊C4×C47—21no—
90C2×C22⋊C4×C47—21no—
91C2×C22⋊C4×C47—21no—
92C2×C22⋊C4×C47D412yes—
93C2×C42×C47—21no—
94C2×C42×C47—21no—
95C2×C42×C47C2×C412yes—
96C24×C47C2×D411yescharacteristic
97C23×D47—21no—
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47—41no—
101C22×C4×C47C22×C411yescharacteristic
102C22×C4×C47C22×C411yescharacteristic
103C22×C4×C47C22×C411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C22×C4×C47#4111yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C22×C4×C47#4111yescharacteristic
113C22×C4×C47#4111yescharacteristic
114C22×C4×C47#4111yescharacteristic
115C2×C22⋊D47—22no—
116C2×C22⋊D47—22no—
117C2×C22⋊D47—41no—
118C2×C22⋊D47—41no—
119C2×C22⋊D47—41no—
120C2×C22⋊D47—41no—
121C22×Dic47—21no—
122C22×Dic47—41no—
123#10625350—22no—
124#10625350—22no—
125#10625350—22no—
126#10625350—22no—
127#10625350—22no—
128#10625350—22no—
129#10625350—22no—
130#10625350—22no—
131#10625350—41no—
132#10625350—41no—
133#10625350—41no—
134#10625350—41no—
135#10625350—41no—
136#10625350—41no—
137#10625350—41no—
138#10625350—41no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188C4212yes—
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—41no—
148C2×C4×D47—41no—
149C2×C4×D47—41no—
150C2×C4×D47—41no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94—21no—
156C2×Dic94C4212yes—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—22no—
159C4⋊C4×C47—22no—
160C4⋊C4×C47—41no—
161C4⋊C4×C47—41no—
162C4⋊C4×C47—41no—
163C22⋊C4×C47—22no—
164C22⋊C4×C47—22no—
165C22⋊C4×C47—41no—
166C22⋊C4×C47—41no—
167C22⋊C4×C47—41no—
168C22⋊C4×C47C22⋊C414yes—
169C42×C47—22no—
170C42×C47—22no—
171C42×C47—22no—
172C42×C47—22no—
173C42×C47—22no—
174C42×C47—41no—
175C22⋊Dic47—22no—
176C22⋊Dic47—41no—
177C4⋊Dic47—22no—
178C4⋊Dic47—41no—
179C4×Dic47—22no—
180C4×Dic47—41no—
181#10625329—22no—
182#10625329—22no—
183#10625329—22no—
184#10625329—22no—
185#10625329—22no—
186#10625329—22no—
187#10625329—22no—
188#10625329C22⋊C414yes—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—22no—
193C2×C47⋊C8—22no—
194C2×C47⋊C8C22⋊C414yes—
195C23×C47—81no—
196C23×C47C2×C22⋊C411yescharacteristic
197C23×C47C4×D411yescharacteristic
198C23×C47C4×D411yescharacteristic
199C23×C47#12111yescharacteristic
200C23×C47#12111yescharacteristic
201C23×C47#12411yescharacteristic
202C23×C47#12711yescharacteristic
203C22×D47—22no—
204C22×D47—41no—
205C22×D47—41no—
206C22×D47—82no—
207C2×C4×C47—24no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—82no—
211C2×C4×C47—82no—
212C2×C4×C47—82no—
213C2×C4×C47C2×C4211yescharacteristic
214C2×C4×C47C2×C22⋊C411yescharacteristic
215C2×C4×C47C2×C22⋊C411yescharacteristic
216C2×C4×C47#11711yescharacteristic
217C2×C4×C47#11711yescharacteristic
218C2×C4×C47#11711yescharacteristic
219C2×C4×C47C4×D411yescharacteristic
220C2×C4×C47C4×D411yescharacteristic
221C2×C4×C47C4×D411yescharacteristic
222C2×C4×C47C4×D411yescharacteristic
223C2×C4×C47C4×D411yescharacteristic
224C2×C4×C47C4×D411yescharacteristic
225C2×C4×C47C4×D411yescharacteristic
226C2×C4×C47C4×D411yescharacteristic
227C2×C4×C47C4×D411yescharacteristic
228C2×C4×C47C4×D411yescharacteristic
229C2×C4×C47#12011yescharacteristic
230C2×C4×C47#12111yescharacteristic
231C2×C4×C47#12111yescharacteristic
232C2×C4×C47#12111yescharacteristic
233C2×C4×C47#12111yescharacteristic
234C2×C4×C47#12311yescharacteristic
235C2×C4×C47#12311yescharacteristic
236C2×C4×C47#12311yescharacteristic
237C2×C4×C47#12411yescharacteristic
238C2×C4×C47#12411yescharacteristic
239C2×C4×C47#12611yescharacteristic
240C2×C4×C47#12611yescharacteristic
241C22⋊D47—42no—
242C22⋊D47—42no—
243C22⋊D47—42no—
244C22⋊D47—42no—
245C2×Dic47—22no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—42no—
249C2×Dic47—82no—
250D188—22no—
251D188—22no—
252D188—41no—
253D188—41no—
254C4×D47—42no—
255C4×D47—42no—
256C4×D47—42no—
257C4×D47—42no—
258Dic94—22no—
259Dic94—22no—
260Dic94—41no—
261Dic94—41no—
262C47⋊C8—24no—
263C47⋊C8—24no—
264#10951—471nomaximal, Sylow
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47—41no—
272C22×C47C4×C22⋊C411yescharacteristic
273C22×C47#38611yescharacteristic
274C22×C47#38811yescharacteristic
275C22×C47#38811yescharacteristic
276C22×C47#39011yescharacteristic
277C22×C47#39011yescharacteristic
278C22×C47#39011yescharacteristic
279C2×D47—42no—
280C2×D47—82no—
281C2×D47—82no—
282C2×D47—82no—
283C4×C47—44no—
284C4×C47—82no—
285C4×C47—82no—
286C4×C47C4×C22⋊C411yescharacteristic
287C4×C47C4×C22⋊C411yescharacteristic
288C4×C47#38611yescharacteristic
289C4×C47#38611yescharacteristic
290C4×C47#38811yescharacteristic
291C4×C47#38811yescharacteristic
292C4×C47#38811yescharacteristic
293C4×C47#38811yescharacteristic
294Dic47—42no—
295Dic47—82no—
296C2×#421—471no—
297#2896—471no—
298C2×#410—471no—
299#2873—471no—
300#1740D4711yescharacteristic, complemented
301#1733—471no—
302C2×H(3,ℤ/4)—471no—
303C2×C47—21no—
304C2×C47—21no—
305C2×C47—21no—
306C2×C47—21no—
307C2×C47—21no—
308C2×C47—21no—
309C2×C47—21no—
310C2×C47#143711yescharacteristic
311D47—82no—
312C22×#41—471no—
313C22×M16—471no—
314C22×M16—471no—
315C2×#117—471no—
316C2×#117C2×D4711yescharacteristic
317C2×#117C2×D4711yescharacteristic
318C2×#117C2×D4711yescharacteristic
319#421—942no—
320#421—942no—
321C2×#104—474no—
322#419—472no—
323#418—472no—
324#417—942no—
325#416—942no—
326#410—474no—
327#410—942no—
328C2×#99—942no—
329#405—942no—
330#405—942no—
331C4×M16—942no—
332#342—22no—
333#339—942no—
334H(3,ℤ/4)—942no—
335C47#1095111yescharacteristic, complemented, Sylow
336C2×#41—472no—
337C2×#41—472no—
338C2×#41—941no—
339C2×#41—941no—
340C2×#41—941no—
341C2×#41—941no—
342C2×#41—941no—
343C2×#41—941no—
344C2×#41—941no—
345C2×#41—941no—
346C2×#41—941no—
347C2×#41—941no—
348C22×Q8—471no—
349C22×D4—471no—
350C22×D4—941no—
351C23×C4—941no—
352C23×C4C22×D4711yescharacteristic
353C2×M16—472no—
354C2×M16—472no—
355C2×M16—472no—
356C2×M16—472no—
357C2×M16—472no—
358C2×M16—941no—
359C2×M16—941no—
360C2×M16—941no—
361C2×M16—941no—
362C2×M16—941no—
363C2×M16—941no—
364C2×M16—941no—
365C22×C8—472no—
366C22×C8—941no—
367#117—21no—
368#117—21no—
369#117—21no—
370#117—21no—
371#117—21no—
372#117—21no—
373#117—21no—
374#117—21no—
375#117—21no—
376#117—472no—
377#117—941no—
378#117—941no—
379#117—941no—
380#117C4×D4712yes—
381#117C4×D4712yes—
382#117D18812yes—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—941no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—941no—
390C2×C22⋊C4C22⋊D4712yes—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—941no—
394C2×C42C4×D4712yes—
395#104—948no—
396#103—944no—
397#102—944no—
398#99—948no—
399#97—944no—
400C4×C8—944no—
401C24—941no—
402C24C2×C22⋊D4711yescharacteristic
403#41—942no—
404#41—942no—
405#41—942no—
406#41—942no—
407#41—942no—
408#41—942no—
409#41—942no—
410#41—942no—
411#41—1881no—
412#41—1881no—
413#41—1881no—
414#41—1881no—
415#41—1881no—
416#41—1881no—
417#41—1881no—
418#41—1881no—
419C2×Q8—472no—
420C2×Q8—941no—
421C2×Q8—941no—
422C2×Q8—941no—
423C2×Q8—941no—
424C2×Q8—941no—
425C2×D4—472no—
426C2×D4—942no—
427C2×D4—942no—
428C2×D4—941no—
429C2×D4—941no—
430C2×D4—941no—
431C2×D4—941no—
432C2×D4—941no—
433C2×D4—1881no—
434C2×D4—1881no—
435C2×D4—1881no—
436C2×D4—1881no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—41no—
440C22×C4—941no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—1881no—
444C22×C4—1881no—
445C22×C4—1881no—
446C22×C4—1881no—
447C22×C4—1881no—
448C22×C4C2×C4×D4711yescharacteristic
449C22×C4C2×C4×D4711yescharacteristic
450C22×C4C2×C4×D4711yescharacteristic
451C22×C4C2×D18811yescharacteristic
452C22×C4#1062535011yescharacteristic
453C22×C4#1062535011yescharacteristic
454C22×C4D4×D4711yescharacteristic
455C22×C4D4×D4711yescharacteristic
456C22×C4D4×D4711yescharacteristic
457C22×C4D4×D4711yescharacteristic
458C22×C4#1062535211yescharacteristic
459C22×C4#1062535211yescharacteristic
460C22×C4#1062535411yescharacteristic
461C22×C4#1062535411yescharacteristic
462M16—474no—
463M16—942no—
464M16—942no—
465M16—942no—
466M16—942no—
467M16—942no—
468M16—942no—
469M16—942no—
470C2×C8—474no—
471C2×C8—942no—
472C2×C8—942no—
473C2×C8—942no—
474C2×C8—942no—
475C2×C8—942no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—942no—
483C4⋊C4—1881no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—942no—
490C22⋊C4—1881no—
491C22⋊C4(C2×D47)⋊C414yes—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—41no—
498C42—942no—
499C42—1881no—
500C23—81no—
501C23—942no—
502C23—3762no—
503C23—1881no—
504C23—1881no—
505C23C2×(C2×D47)⋊C411yescharacteristic
506C23C4×C22⋊D4711yescharacteristic
507C23C4×C22⋊D4711yescharacteristic
508C23#1309399611yescharacteristic
509C23#1309399611yescharacteristic
510C23#1309401311yescharacteristic
511C23#1309402011yescharacteristic
512Q8—942no—
513Q8—942no—
514Q8—1881no—
515Q8—1881no—
516D4—942no—
517D4—942no—
518D4—1882no—
519D4—1882no—
520D4—1882no—
521D4—1882no—
522D4—1881no—
523D4—1881no—
524C2×C4—24no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—942no—
531C2×C4—3762no—
532C2×C4—1882no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1882no—
536C2×C4—1882no—
537C2×C4—1882no—
538C2×C4—1882no—
539C2×C4C42×D4711yescharacteristic
540C2×C4C22⋊C4×D4711yescharacteristic
541C2×C4C22⋊C4×D4711yescharacteristic
542C2×C4#1309398111yescharacteristic
543C2×C4#1309398211yescharacteristic
544C2×C4#1309398211yescharacteristic
545C2×C4(C22⋊D47)⋊C411yescharacteristic
546C2×C4(C22⋊D47)⋊C411yescharacteristic
547C2×C4(C22⋊D47)⋊C411yescharacteristic
548C2×C4(C22⋊D47)⋊C411yescharacteristic
549C2×C4#1309398611yescharacteristic
550C2×C4#1309398611yescharacteristic
551C2×C4#1309398611yescharacteristic
552C2×C4#1309398611yescharacteristic
553C2×C4C4×D18811yescharacteristic
554C2×C4C4×D18811yescharacteristic
555C2×C4#1309399411yescharacteristic
556C2×C4#1309399711yescharacteristic
557C2×C4#1309399711yescharacteristic
558C2×C4#1309399911yescharacteristic
559C2×C4#1309399911yescharacteristic
560C2×C4#1309400811yescharacteristic
561C2×C4#1309400911yescharacteristic
562C2×C4#1309400911yescharacteristic
563C2×C4#1309401011yescharacteristic
564C2×C4#1309401011yescharacteristic
565C2×C4#1309401911yescharacteristic
566C2×C4#1309401911yescharacteristic
567C8—944no—
568C8—944no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—3762no—
577C22—3762no—
578C22—3762no—
579C22—1882no—
580C22C4×(C2×D47)⋊C411yescharacteristic
581C22#1568900011yescharacteristic
582C22#1568901311yescharacteristic
583C22#1568901311yescharacteristic
584C22#1568902311yescharacteristic
585C22#1568902311yescharacteristic
586C22#1568902411yescharacteristic
587C4—44no—
588C4—82no—
589C4—82no—
590C4—3762no—
591C4—1882no—
592C4#1568897011yescharacteristic
593C4#1568897011yescharacteristic
594C4#1568900111yescharacteristic
595C4#1568900111yescharacteristic
596C4#1568901211yescharacteristic
597C4#1568901211yescharacteristic
598C4#1568901511yescharacteristic
599C4#1568901511yescharacteristic
600C2—21no—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—3762no—
608C2#2024162411yescharacteristic
subgroupG/H
centerC4#15689001
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC47×#1740C2
socleC2×C47#1437
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenter#1740D47
hypocenterC47#10951
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1740

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #20251807C2semidirect
5. #20243098C2semidirect
6. C2×#15688879C2semidirect
7. C47×#1740C2semidirect
14. C2×#13093952C4semidirect
15. #15689131C4semidirect
16. #15689127C4semidirect
20. #15689096C4semidirect
300. #1740D47semidirect
335. C47#10951semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156891412
#202519052
C2×#156890962
#202518072
#202430982
C2×#156888792
C47×#17402
#1095147